Global counterexamples to uniqueness for a Calderón problem with C^k conductivities

2024/06/20 by Thierry Daudé, Bernard Helffer, Daudé, Thierry +5 · 1 citation
Computer Science · Engineering · Mathematics · #35P25 #58J50 #81U40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Boundary value problem #Bounded function #Combinatorics #Counterexample #Dirichlet distribution #Domain (mathematical analysis) #FOS: Mathematics #FOS: Physical sciences #Lambda #Mathematical Physics (math-ph) #Mathematical analysis #Mathematics #Nabla symbol #Numerical methods in inverse problems #Omega #Physics #Quantum mechanics #Spectral Theory (math.SP) #Stability and Controllability of Differential Equations #Uniqueness

paper · pdf · doi:10.48550/arxiv.2406.14063

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2024/06/20 · openalex created_date 2024/06/22 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ Rⁿ, n ≥ 3, be a fixed smooth bounded domain, and let γ be a smooth conductivity in Ω. Consider a non-zero frequency λ₀ which does not belong to the Dirichlet spectrum of L_γ= -\rm div (γ∇ ⋅). Then, for all k ≥ 1, there exists an infinite number of pairs of non-isometric C^k conductivities (γ₁, γ₂) on Ω, which are close to γ such that the associated DN maps at frequency λ₀ satisfy Λ_γ1,λ0 = Λ_γ2,λ0.

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• The Dirichlet-to-Neumann map on asymptotically anti-de Sitter spaces and holography

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